Protected qubit based on superconducting current mirror

ABSTRACT

A qubit implementation based on exciton condensation in capacitively coupled Josephson junction chains is disclosed. The qubit may be protected in the sense that unwanted terms in its effective Hamiltonian may be exponentially suppressed as the chain length increases. Also disclosed is an implementation of a universal set of quantum gates, most of which offer exponential error suppression.

BACKGROUND

Physical implementation of a quantum computer presents a great challengebecause quantum systems are susceptible to decoherence and becauseinteractions between them cannot be controlled precisely. Quantum bits(a.k.a., qubits) must satisfy two basic requirements: they must preservethe quantum state intact for a sufficiently long time, and they must beeasily operable. It has proved very difficult to meet both conditionstogether.

There have been impressive demonstrations of qubits using various kindsof systems, including Josephson junctions, yet building a full-scalecomputer remains a remote goal. In principle, scalability can beachieved by correcting errors at the logical level, but only if thephysical error rate is sufficiently small. As an alternative solution,it has been observed that topologically ordered quantum systems arephysical analogues of quantum error-correcting codes, and fault-tolerantquantum computation can be performed by braiding anyons.

Simpler examples of physical systems with error-correcting propertieshave been found. The key element of such systems, which may be referredto herein as 0-π qubit, is a two-terminal circuit built of Josephsonjunctions. Its energy has two equal minima when the superconductingphase difference between the terminals, θ=φ₁−φ₂, is equal to 0 or π. Thequantum states associated with the minima, |0> and |1> can form quantumsuperpositions. It is essential that the energy difference between thetwo minima is exponentially small in the system size, even in thepresence of various perturbations, hence the quantum superposition willremain unchanged for a long time. Implementations of some quantum gateshave also been proposed.

SUMMARY

The application of qubits for universal or specialized quantumcomputation requires some implementation of quantum gates. The followinggates are described: 1) measurement in the standard basis; 2)measurement in the dual basis; 3) one-qubit unitary gateR(π/4)=exp(i(π/4)σ^(z) and its inverse Josephson junction currentmirror; 4) two-qubit unitary gate R₂(π/4)=exp(i(π/4)σ₁ ^(z) ₂ ^(z) andits inverse; and 5) one-qubit unitary gate R(π/8)=exp(i(π/8)σ^(z) andits inverse.

It is very desirable for the gates to be fault-tolerant at the physicallevel or at least to limit possible errors to some particular types.Such an implementation is described herein. It involves a choice of anonstandard but computationally universal set of gates that areparticularly suitable for use with 0-π qubits. The computation isadaptive, i.e., it involves intermediate measurements whose outcomedetermines the choice of the next gate to be applied. A concreterealization of the gates is described.

To show that the set of gates disclosed herein is universal, it mayfirst be observed that repeated applications of noncommutingmeasurements allow one to prepare any of these states: |0>, |1>, |+>,|−>. It may also be observed that R(−π/4) is equal to Λ(i) up to anoverall phase, where Λ(i)|a>=i^(a)|a> for a=0, 1. One can also implementthe two-qubit controlled phase gate Λ²(−1) that acts as follows:Λ²(−1)|a,b>=(−1)^(ab)|a,b>. Specifically, Λ²(−1) is equal toR₂(−π/4)(R(−π/4){circle around (x)}R(−π/4)) up to an overall phase. Ifwe add the Hadamard gate H, we obtain all Clifford (i.e., simplectic)gates.

The Hadamard gate may be realized according to the following adaptiveprocedure. A qubit may be provided in an arbitrary state|ψ>=c₀|0>+c₁|1>, and supplemented with a |+> ancilla. Then, Λ²(−1) maybe applied. The first qubit may then be measured in the dual basis. Thesecond qubit now contains H|ψ> or σ^(x)H|ψ>, depending on themeasurement outcome. In the second case, the procedure may be repeated2, 4, 6, . . . times, until a desired result is achieved.

Using the Clifford gates and the ability to create copies of|ξ>=R(π/8)|+>, one can perform quantum computation. Furthermore, if theClifford gates are exact, the ancillary state |ξ> needs to be preparedwith fidelity F>0.93. This gives more than 50% tolerance for choosingthe parameter u≈π/8 in R(u)=exp(iuσ^(z)).

Thus, a concrete design that belongs to a class of 0-π superconductingqubits may be described. Such a design may include a current mirrordevice with four leads connected diagonally. This design makes itpossible to perform a measurement with respect to the dual basis bybreaking one of the connecting wires, using whatever technique that issuitable to measure the offset charge of a capacitor. A fault-tolerantscheme is also described, including a universal set of gates and theirschematic implementations. This scheme can be used with any kind of 0-πsuperconducting qubits.

A qubit may be implemented as a circuit of Josephson junctions withbuilt-in error-correcting properties. As a result, the rate ofdecoherence (decay of the quantum state) may be decreased by a factorthat is exponential in the circuit size. Quantum operations can beperformed by breaking some connections in the circuit, or by connectingtwo qubits via superconducting wires. Some aspects of the quantumprotection remain in effect even when the operations are performed.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a functional block diagram of an example superconductingcurrent mirror.

FIGS. 1B and 1C depicts a physical embodiment of a portion of thesuperconducting mirror of FIG. 1A.

FIG. 2 depicts measurement in the standard basis.

FIG. 3 depicts measurement in the dual basis.

FIG. 4 depicts an example realization of the gate R(u) for arbitrary u.

FIGS. 5A and 5B depict example fault-tolerant realizations of R(π/4) andR₂(π/4), respectively.

DETAILED DESCRIPTION

A novel design of a 0-π qubit based on the current mirror effect incapacitively coupled chains of Josephson junctions will now bedescribed. An analogue of this effect in normal-metal junctions is dueto correlated electron-hole tunneling. In superconducting chains, thetunneling objects are Cooper pairs. Positive and negative Cooper pairs(with electric charge +2e in one chain and −2e in the other chain) tendto tunnel together. Under suitable conditions, the currents in the twochains are opposite in direction and almost equal in magnitude. This hasbeen observed experimentally in the resistive state, i.e., atsufficiently large voltage bias. A more delicate, dissipationless formof this effect, which has not been observed, has been predictedtheoretically for the case of strong interchain coupling. In thisregime, the Josephson junction ladder behaves as an almost perfect DCtransformer with 1:1 current-to-voltage ratio.

FIG. 1A is a functional block diagram of an example superconductingcurrent mirror, or “ladder.” As shown, a current mirror may include aplurality of Josephson junctions J, interconnected by superconductingwires W. Each Josephson junction may have Josephson energy J andcapacitance C₁. Interchain capacitors C (connected to form “rungs” onthe “ladder”) may have capacitance C₂. The mirror may have fourterminals 1-4.

If C₂>>C₁, excitons of +2e in one chain and −2e in the other chain mayhave lower energy than individual ±2e quasiparticles or otherexcitations that change the total charge on some rungs of the ladder.The energy scales for excitons and unbalanced charge fluctuations aregiven by E_(ex)˜e²/C₂ and E₁˜e²/C₁, respectively. Excitons form a Bosecondensate if E_(ex){tilde under (<)}J_(ex), where J_(ex)˜J²/E₁ is acharacteristic hopping energy (we assume J{tilde under (<)}E₁). In thisregime, the system becomes superconducting with respect to oppositecurrents in the two chains while being insulating with respect topassing net electric charge along the ladder. It is worth noting thatthe exciton condensate persists in the presence of charge frustration.

The current mirror device may be characterized by an effective potentialenergy E that depends on the values of the superconducting phase φ₁, φ₂,φ₃, φ₄ at each of the four terminals 1-4, respectively. The orderparameter of the exciton condensate may be represented as thesuperconducting phase difference between the chains, which is equal toθ₁=φ₁−φ₂ at the left end of the ladder and θ_(r)=φ₄−φ₃ at the right endof the ladder. Thus, the energy may be expected to depend primarily onθ₁−θ_(r)=φ₁−φ₂+φ₃−φ₄:

E=F(φ₁−φ₂+φ₃−φ₄)+φ₁−φ₄, φ₂−φ₃),  (1)

where f is an “error term.” Since the current through the j-th terminalis proportional to ∂E/∂φ, the error term characterizes the net currentthrough the ladder. Such current can only be carried by ±2equasiparticles tunneling through the insulator, but this process issuppressed by factor exp(—N/N₀), wherein N is the length (i.e., thenumber of junctions in each chain) and N₀˜1. On the other hand, the Fterm in Eq. (1) is of the order of J_(ex)/N.

Such a current mirror may be realized physically as depicted in FIGS. 1Band 1C. As shown in FIG. 1C, a layer of Al₂O₃, for example, may bedeposited on a first layer of Al. A second layer of Al may be depositedover the layer of Al₂O₃. As best seen in FIG. 1B, the amount that thesecond Al layer overlaps the Al₂O₃ layer determines the amount ofcapacitance, while the thickness of the Al₂O₃ layer determines whethertunneling will occur. Thus, as shown, with a relatively small overlapand a thin Al₂O₃ layer, a Josephson junction J may be formed. Where theoverlap is relatively large but the insulating Al₂O₃ layer is thicker, aclassical capacitor may be formed.

A simple design of the 0-π qubit will now be described. The four leads1-4 may be connected diagonally (i.e., 1 with 3 and 2 with 4). Thus,φ₁=φ₃, φ₂=φ₄, and E≈F(2(φ₁−φ₂)) with exponential precision. As thefunction F(θ) has a minimum at θ=0, the energy of the qubit has twominima, at φ₁−φ₂=0 and φ₁−φ₂=π. Note that all variables φ_(j) aredefined modulo 2π. The energy values at the minima are exponentiallyclose to each other: δE∝exp(−N/N₀). That is the reason for protectionagainst dephasing. To prevent bit flips, one needs to make sure thatE>>e²/C, where C=NC₂ is total interchain capacitance. Note that theratio E/(e²/2C)˜J_(ex)/E_(ex) does not depend on length. It can beincreased by increasing the interchain coupling or by connecting severalcurrent mirrors in parallel.

With this qubit design, which saves the quantum state of the qubit, itis possible to do measurements in the standard bases (of states |0> and|1> corresponding to φ₁−φ₂=0 and φ₁−φ₂=π, respectively) as well as thedual basis,

$ {{{{ \pm}\rangle} = {{{\frac{1}{\sqrt{2}}( {0}\rangle } \pm}1}}\rangle} ).$

They may also be called “phase basis” and “charge basis,” respectively.

Measurement in the phase basis, as shown in FIG. 2, may be performed byconnecting leads 1 and 2 to a measuring current. For example, if theleads are connected via a Josephson junction J, the current I in theloop (from terminal 2 to terminal 1 through the Josephson junction J)depends on φ₁−φ₂=0 and the magnetic flux 4 through the loop, i.e., I=Jsin (φ₁−φ₂+Φ). The state (e.g., |0> or |1>) can be determined from thedirection in which the current I flows. It should be understood that,although only a single 0-π qubit is shown in FIG. 2, measurement in thestandard (phase) basis may employ any system of one or more 0-π qubits.

Measurement in the dual basis may be more complicated. As shown in FIG.3, the key idea is to break the wire between 1 and 3, and attach someoffset charge measuring (OCM) circuit to those leads. Now

φ₂=φ₄≡(φ₁+φ₃)/2(modπ).  (2)

It should be understood that current will not flow, but the currentmirror will charge. Furthermore, the potential energy is practicallyindependent of the superconducting phase difference, θ=φ₁−φ₃ across thedevice. Consequently, direct superconducting current cannot flow. Thedevice behaves basically as a capacitor with the effective Hamiltonian

$\begin{matrix}{{H_{cap} = {\frac{( {2e} )^{2}}{2C}( {\frac{\partial}{i{\partial\theta}} - n_{g}} )^{2}}},} & (3)\end{matrix}$

except that it has an internal degree of freedom, because, for fixedvalues of φ₁, φ₃, Eq. (2) has two solutions. The states |+> and |−>correspond to the symmetric and antisymmetric superposition of thesesolutions, and the wave function ψ(θ) satisfies the boundary conditionψ(2π)=ψ(0) or ψ(2π)=−ψ(0), respectively. The second boundary conditionbecomes equivalent to the first one if n_(g) is changed by ½. Theparameter n_(g) is a so-called offset charge measured in units of 2e. Itis defined modulo 1. Thus, the measurement in the |±> basis amounts todistinguishing between n_(g) and n_(g)+½. From a practical perspective,n_(g) need not be known in advance. Indeed, it is only important to tellthe two states apart while the labels “+” and “−” can be assignedarbitrarily.

Offset charge corresponds to the internal state of qubit. Accordinglywhen offset charge is measured, the internal state of the qubit ismeasured. Charge, however, cannot be measured directly. The OCM circuitmeasures the difference between energy levels of the system. The energyspectrum of a capacitor is described by the formulaE_(n)=(2e²/C)(n−n_(g))², hence the value of n_(g) can be inferred fromthe measurement of the difference between two levels, e.g., E₁ and E₀.Where n_(g)=0, the state is |+>. Where n_(g)=½, the state is |−>.

It can readily be seen that the implementation of measurements describedabove is fault-tolerant since the measured observable (i.e., thesuperconducting phase or the offset charge) is as unlikely to changeduring the process as in an isolated qubit.

A one-qubit unitary gate R(π/8)=exp(i(π/8)σ^(z)), and its inverse, willnow be described with reference to FIG. 4. Generally, a gate R(u) may berealized by connecting the leads 1 and 2 to a Josephson junction J for acertain period of time. This procedure is generally sensitive to randomvariations of the time interval and the strength of the Josephsoncoupling.

To operate the gate, the switch may be closed for an interval of time.While the switch is closed, the qubit state evolves in a certain way.The effect of the switch closure is that the qubit state is multipliedby the operator R(π/8). Accordingly, such a gate may be used to changethe state of the system in a certain way.

A circuit for a fault-tolerant implementation of R(π/4)=√{right arrowover (i)}Λ(−i) is shown schematically in FIG. 5A. The 0-π qubit isconnected to an ultraquantum LC-oscillator (with

$r\overset{def}{=}( {{{^{2}/h}\sqrt{L/C}}1} )$

for a certain period of time τ. The operation of this gate may bedescribed in terms of the superconducting phase difference θ across theinductor.

Initially, the oscillator is in its ground state characterized by aGaussian wave function ω₀(θ). Note that

θ²

˜r>

1. Once the circuit is closed, the quantum evolution is governed by theeffective Hamiltonian

H _(L)=(h ²/8e ²)L ⁻¹θ²  (4)

where θ takes on multiples of 2π if the qubit is in the state 0, or onvalues of the form 2π(n+½) if the qubit is in the state 1. Thus, thewave function ψ(θ) has the form of a grid: it consists of narrow peaksat the said locations. If τ=8L(e²/h), then all peaks with θ=2πn pick upno phase and all peaks with θ=2π(n+½) are multiplied by −i. Thus, thegate Λ(−i) is effectively applied to the qubit state, not entangling itwith the oscillator.

In other words, when the switch is closed, a superposition of states isobtained, each state having a different current through the inductor.While the switch is closed, the superposition evolves. After time, thesuperposition assumes a certain form, such that when the switch isopened, an effective operation has been applied to the qubit.

Such a gate may be operated in three phases. First, the switch may beclosed. Then, the superposition may be allowed to evolve for an intervalof time. Third, the system may be transitioned back to its qubit state(i.e., the switch may be opened). Then, one can measure the state of thesystem as described above. This gate tolerates small errors in the timeit takes to open and close the switch.

Some conditions should be met for this scheme to work. The closing andbreaking of the circuit should occur smoothly enough so that noexcitation is produced in the switch itself, but faster than the LCoscillation period. Then the qubit is transformed into a Gaussian gridstate, i.e., the superposition of peaks with a Gaussian envelope. Suchstates are known to have good error-correcting properties. If theprotocol is not followed exactly, but with small error, the error willmainly result in oscillations in the LC circuit after the cycle iscomplete, leaving the qubit state unaffected.

As shown in FIG. 5B, the gate R₂(π/4) is implemented similarly. Onemerely needs to connect the two 0-π qubits in series. This gate involvesinteraction between qubits. A quantum computer may be defined by asystem of qubits. One needs to perform certain transformations withthese qubits. The state of the system of qubits can be evolved usingthis gate.

1. A qubit circuit for a quantum computer, the qubit circuit comprisinga Josephson junction current mirror.
 2. The qubit circuit of claim 1,further comprising: first, second, third, and fourth terminals coupledto the Josephson junction current mirror, wherein the first and thirdterminals are current input terminals and the second and fourthterminals are current output terminals.
 3. The qubit circuit of claim 2,wherein the terminals are coupled to the current mirror viasuperconducting wires.
 4. The qubit circuit of claim 2, wherein thesecond terminal is connected to the fourth terminal via asuperconducting wire.
 5. The qubit circuit of claim 4, wherein the firstterminal is connected to the third terminal via a superconducting wire.6. The qubit circuit of claim 5, wherein the first and fourth terminalsare electrically connected to a first series of Josephson junctions, andthe second and third terminals are electrically connected to a secondseries of Josephson junctions.
 7. The qubit circuit of claim 1, whereinthe current mirror comprises a first series of Josephson junctions and asecond series of Josephson junctions, and each point between twoadjacent Josephson junctions in the first series is capacitively coupledto a corresponding point between two adjacent Josephson junctions in thesecond series.
 8. The qubit circuit of claim 7, further comprising afirst terminal coupled to the first series of Josephson junctions, asecond terminal coupled to the second series of Josephson junctions, athird terminal coupled to the second series of Josephson junctions, anda fourth terminal coupled to the first series of Josephson junctions. 9.The qubit circuit of claim 8, further comprising a first superconductingwire connecting the second and fourth terminals.
 10. The qubit circuitof claim 9, further comprising a second superconducting wire connectingthe first and third terminals.
 11. The qubit circuit of claim 7, whereinthe current mirror comprises a plurality of capacitors, each of which iscoupled to a first pair of Josephson junctions from the first series andto a second pair of Josephson junctions from the second series.
 12. Alogical gate for a quantum computer, the logical gate comprising: apartial qubit, the partial qubit comprising: a Josephson junctioncurrent mirror; and first, second, third, and fourth terminals coupledto the Josephson junction current mirror, wherein the second terminal isconnected to the fourth terminal via a superconducting wire; and acircuit that is electrically coupled to the partial qubit, the circuitbeing adapted to provide for a measurement of a quantum state of thepartial qubit.
 13. The logical gate of claim 12, wherein the circuitcomprises an offset charge measuring circuit coupled to the first andthird terminals.
 14. The logical gate of claim 13, wherein the offsetcharge measuring circuit provides for measurement of a dual basisquantum state of the partial qubit.
 15. The logical gate of claim 14,wherein the offset charge measuring circuit measures an energy spectrumof the current mirror.
 16. A logical gate for a quantum computer, thelogical gate comprising: a qubit with two external superconductingterminals, wherein standard basis states |0> and |1> are characterizedby a superconducting phase difference 0 and π between the terminals; andan inductor-capacitor circuit coupled to the qubit, wherein the circuitis adapted to affect a quantum superposition of the standard basisstates.
 17. The logical gate of claim 16, wherein the qubit comprises aJosephson junction current mirror with four terminals, wherein two ofthe four terminals are current input terminals that are connected to oneanother via a first superconducting wire, and two of the four terminalsare current output terminals that are connected to one another via asecond superconducting wire, and wherein the two wires are the externalsuperconducting terminals of the qubit.
 18. The logical gate of claim17, wherein the circuit comprises a switch disposed between theinductor-capacitor circuit and the first qubit.
 19. The logical gate ofclaim 18, wherein the inductor-capacitor circuit is coupled to one ofthe current output terminals, and the switch is disposed between theinductor-capacitor circuit and one of the current input terminals. 20.The logical gate of claim 19, further comprising a second qubit coupledto the inductor-capacitor circuit in series with the first qubit.